21 thoughts on “Permutations and Combinations – 5 Card Poker Hands

  1. thank you very much this is a good explanation. had a problem in ap stats
    asking to explain the probablility of 3 of a kind and you explained it
    perfectly. thanks again

  2. To give some more examples, (13C3) means to grab any three ranks from the
    13 available ranks. So (13C3)(4C2)(4C1)(4C3) means grab any three ranks,
    then choose 2 of the 4 cards for the first rank, then 1 of the 4 cards for
    the second rank, and 3 of the 4 cards from the third rank. So how you set
    it up depends on what you’re hand looks like. That’s why your original
    examples were set up differently. Just to make sure, rank means a type of
    card. Ace is a rank, 2 is a rank, 3 is a rank, etc.

  3. 13C1 is just used to represent you choosing one rank (type of card). But it
    could be any rank. For example, (13C1) means it could be K, 10, Ace, 5,
    Queen, 2, etc. So (13C1)(4C1) means to choose a rank, and then one card
    from that rank. (13C1)(4C2) it means a pair from any rank. (13C1)(4C3)
    means choose a rank and then choose 3 cards from that rank. But (13C1)
    isn’t always used. For example, two pair uses (13C2)(4C2)(4C2). (13C2)
    means pick any two ranks from the 13 available ranks.

  4. Thank you so much for the clearing, I thought I loved math! I feel like a
    child when it comes to permutation and combinations. So, 13C1 could only be
    used if we are asked for a pair, 2 pairs, 3 or four of a kind, full house,
    and royal flush?

  5. The second example is a very specific type of 3 of a kind. It won’t have
    the (13C1) because your 3 of a kind are JACKS. Plus your 4th card is fixed
    as a Queen. This is why there is no (12C2) anymore from my example. So
    (4C1) for the queen, (4C3) for the 3 Jacks, (44C1) to find the last card
    from the remaining 44 ranks that is not a Queen or Jack. Hope this helps.
    Just remember your examples are not general 3 of a kinds. Your formulas for
    them will change because some cards are fixed.

  6. These two examples don’t fit the general three of a kind scenario. Exactly
    three kings means the other two cards could be the same (but not Kings).
    This means you’re including some combinations that are a full house. Plus
    since your 3 of a kind are KINGS, you don’t have the (13C1) at the
    beginning. (13C1) allows for the 3 of a kind to be any card (3 Kings, 3
    10s, 3 Aces, etc).

  7. Hi Brian, my book solves (how many different 5-hand cards are possible that
    consist of exactly 3 kings?) as (4C3)(48C2)=4512. Or, (how many different
    5-hand cards are possible that consist of 1 Queen and 3 Jacks?) as
    (4C1)(4C3)(44C1)=704 How are these different than your way of solving three
    of a kind @ 9:07

  8. It’s the most common mistake, one I made the first time tried to do this
    problem. If you calculate it your way, you’ll get twice as many
    combinations. It’s because of the (13C1)*(12C1) part. If I remember
    correctly, this means the order of the two ranks matter. (KK994 is
    different than 99KK4) We know they are considered the same hand. It’s why
    your number is twice as much as it should be. Instead, use (13C2). This
    allows you to select 2 ranks, where order does not matter. KK994 = 99KK4

  9. I tried the two pair example before he said the answer and I got
    13C1*4C2*12C1*4C2*11C1*4C1. Why is that wrong? Or, what do I misunderstand
    if I got that answer?

  10. 12C1 x 4C1 x 11C1 x 4C1 = 48 x 44 is more like a permutation because if you
    switch the last two cards, this will treat it as a new hand. For example,
    the way you have it written, QQQ5J would be a different hand than QQQJ5. So
    you have to take 12C1 x 4C1 x 11C1 x 4C1 and divide by 2 to account for the
    order of the last two cards. (12C1) x (11C1) / 2 = (48 x 44) / 2 is the
    same as 12C2.

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